A new energy bound for Einstein-Scalar theory in AlAdS${}_4$ and holographic bound for deformed CFT${}_3$

In this work, we derive an upper bound on energetic quantities, namely vacuum energy and free energy, for static solutions of Einstein-Scalar theory in four dimensional asymptotically locally Anti-de Sitter(AlAdS) spacetime with a nontrivial scalar potential where the scalar field mass parameter($m^2$) is equal to 0 or -2. This system is the holographic dual of strongly coupled conformal field theory(CFT) in three dimensions being deformed by a relevant or marginal scalar operator of conformal dimension $\mathrm{\Delta }=1,2,3$. The bound is derived from a purely gravitational perspective regardless of the inhomogeneity of the static conformal boundary of AlAdS and the source of the deformation. We demonstrate the bound in simple settings and check the consistency with the known previous bounds.